Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{7k^2 - 35k - 42}{-4k^2 - 24k - 20}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {7(k^2 - 5k - 6)} {-4(k^2 + 6k + 5)} $ $ p = -\dfrac{7}{4} \cdot \dfrac{k^2 - 5k - 6}{k^2 + 6k + 5} $ Next factor the numerator and denominator. $ p = - \dfrac{7}{4} \cdot \dfrac{(k + 1)(k - 6)}{(k + 1)(k + 5)}$ Assuming $k \neq -1$ , we can cancel the $k + 1$ $ p = - \dfrac{7}{4} \cdot \dfrac{k - 6}{k + 5}$ Therefore: $ p = \dfrac{ -7(k - 6)}{ 4(k + 5)}$, $k \neq -1$